#6431712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(9,10,14,18)(11,19,22,32)(12,16,23,30)(13,25,27,17)(15,26,29,21)(20,33,34,40)(24,37,36,31)(28,38,39,35);(5,6)(7,8)(9,11,14,22)(10,15,18,29)(12,20,23,34)(13,26,27,21)(16,28,30,39)(17,32,25,19)(24,38,36,35)(31,40,37,33);(2,4)(6,8);(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,31)(18,30)(19,33)(21,35)(22,34)(25,37)(26,38)(27,36)(29,39)(32,40);(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338908);